Modified Rayleigh Conjecture Method and Its Applications

0.32 MB·English

by  A. G. Ramm

Modified Rayleigh Conjecture Method and Its
6
0
0 Applications
2

n
Alexander G. RAMM∗
a
J and

2 Semion GUTMAN†
1

]
A
Abstract
N
The Rayleigh conjecture about convergence up to the boundary of
.
h the series representing the scattered field in the exterior of an obstacle
at D is widely used by engineers in applications. However this conjecture
m is false for some obstacles. AGR introduced the Modified Rayleigh Con-
jecture (MRC), which is an exact mathematical result. In this paper we
[
review thetheoretical basis for theMRC methodfor 2D and3D obstacle

1 scattering problems, for static problems, and for scattering by periodic
v structures. Wealso present successful numerical algorithms based on the
8 MRCforvariousscatteringproblems. TheMRCmethodiseasytoimple-
9 ment for both simple and complex geometries. It is shown to be a viable
2
alternative for other obstacle scattering methods. Various direct and in-
1
verse scattering problems require finding global minima of functions of
0
several variables. TheStabilityIndexMethod (SIM)combinesstochastic
6
and deterministic method toaccomplish such a minimization.
0
/ Key words: obstacle scattering, Modified Rayleigh Conjecture, Stability
h
t Index Method.
a
m
Math. Subj. classification: 35J05,65M99, 78A40
:
v
i 1 Introduction
X
r
a In this paper we review our recentresults onthe Modified RayleighConjecture
(MRC)method. Themethodisappliedtomultidimensionalobstaclescattering
problems, as well as to scattering by periodic structures. Also we discuss an
application of the MRC to static problems, and preliminary results on inverse
obstaclescatteringbyMRC.Numericalresultsillustratetheperformanceofvar-
iousMRC algorithms. The paperconcludes withapresentationofthe Stability
Index Method (SIM) for global minimization.
∗Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602,
USA,e-mail: [email protected]
†DepartmentofMathematics,UniversityofOklahoma,Norman,OK73019, USA,e-mail:
[email protected]
1
The basic theoretical foundation of the method was developed in [27]. The
MRC has the appeal of an easy implementation for obstacles of complicated
geometry, e.g. having edges and corners. In our numerical experiments the
methodhasshownitselftobeacompetitivealternativetotheBIEM(boundary
integralequations method), see [13]. Also, unlike the BIEM, one can apply the
algorithm to different obstacles with very little additional effort. A similar
method is discussed in [10]
Weformulatetheobstaclescatteringproblemina3DsettingwiththeDirich-
let boundary condition, but the method can also be used for the Neumann
boundarycondition,correspondingtoacousticallyhardobstacles,andtheRobin
boundary condition.
Consider a bounded domain D R3, with a Lipschitz boundary S. Denote
the exterior domainby D′ =R3 D.⊂Let α,α′ S2 be unit vectors,where S2 is
the unit sphere in R3. \ ∈
The acoustic wave scattering problem by an acoustically soft obstacle D
consists in finding the (unique) solution to the problem (1.1)-(1.2):
2+k2 u=0 in D′, u=0 on S, (1.1)
∇
(cid:0) (cid:1)
eikr 1 x
u=u +A(α′,α) +o , r := x , α′ := . (1.2)
0
r r | |→∞ r
(cid:18) (cid:19)
Hereu :=eikα·x isthe incidentfield,v :=u u is the scatteredfield,A(α′,α)
0 0
−
is called the scattering amplitude, its k-dependence is not shown, k > 0 is the
wavenumber. The scattered field v is an outgoing solution of the Helmholtz
differential equation (1.1), that is, a solution which satisfies the radiation con-
dition
2
∂v
lim ikv ds=0. (1.3)
r→∞ ∂ x −
Z|x|=r(cid:12) | | (cid:12)
(cid:12) (cid:12)
Denote (cid:12) (cid:12)
(cid:12) (cid:12)
A (α):= A(α′,α)Y (α′)dα′, (1.4)
ℓ ℓ
ZS2
where Y (α) are the orthonormal spherical harmonics, Y =Y , ℓ m ℓ.
ℓ ℓ ℓm
− ≤ ≤
Let a ball B := x : x R contain the obstacle D. Let h (r) be the
R ℓ
{ | | ≤ }
sphericalHankel functions, normalized so that h (r) eikr as r + . In the
ℓ ∼ r → ∞
region r >R the solution to (1.1)-(1.2) is:
∞
x
u(x,α)=eikα·x+ A (α)Ψ (x), Ψ (x):=Y (α′)h (kr), r >R, α′ = ,
ℓ ℓ ℓ ℓ ℓ
r
ℓ=0
X
(1.5)
wherer = x, the sumincludesthe summationwithrespectto m, ℓ m ℓ,
| | − ≤ ≤
and A (α) are defined in (1.4), see [26].
ℓ
The Rayleigh conjecture (RC) is: the series (1.5) converges up to the
boundary S (originally RC dealt with periodic structures, gratings). This con-
jectureisfalseformanyobstacles,butistrueforsome([3,22,28]). Forexample,
2
if n=2 and D is an ellipse, then the series analogous to (1.5) converges in the
region x > a, where 2a is the distance between the foci of the ellipse [3]. In
| |
the engineeringliteraturethere arenumericalalgorithmsbasedonthe Rayleigh
conjecture. These algorithms use projection methods and are reported to be
unstable. Moreover, no error estimate has been obtained for such algorithms.
These algorithmscannotconvergefor arbitraryobstacles,because the Rayleigh
conjecture is false for some obstacles.
Our aim is to give a formulation of a Modified Rayleigh Conjecture (MRC)
which holds for any Lipschitz obstacle and canbe used in numericalsolutionof
directandinversescatteringproblems. In otherwords,while the MRC still has
the word ”conjecture” in its name, it is a proven mathematical result for the
scatteredfieldintheexteriordomainD′. Incontrasttoalgorithmsbasedonthe
invalidRayleighConjecture,theMRC-basedalgorithms,liketheonesdescribed
here,convergeandanerrorestimatefortheapproximatesolutiontheyyieldhas
been obtained in [27] (see also [34], Chapter 12). This error estimate is sharp
in the order ǫ.
2 Modified Rayleigh conjecture
WhatwecalltheModifiedRayleighConjecture(MRC)isactuallythefollowing
Theorems2.1,see[27],and2.3,see[16]. WedenotebyHm(D′)thesetoffunc-
loc
tions from the Sobolev space Hm(D˜) for any compactstrictly inner subdomain
D˜ of D′, so that the distance from D˜ to S is positive, dist(D˜,S)>0.
Theorem 2.1. Let v =u u be the scattered field, where u is the solution to
0
−
(1.1)-(1.2). Then there exists a positive integer L = L(ǫ) and the coefficients
c =c (ǫ), 0 ℓ L(ǫ) such that
ℓ ℓ
≤ ≤
(i).
u +v ǫ, (2.1)
0 ǫ L2(S)
|| || ≤
where
L(ǫ)
v (x)= c (ǫ)Ψ (x). (2.2)
ǫ ℓ ℓ
ℓ=0
X
(ii).
v v ǫ (2.3)
ǫ L2(S)
k − k ≤
and
v v =O(ǫ), ǫ 0, (2.4)
ǫ
||| − ||| →
where
|||·|||=k·kHlmoc(D′)+k·kL2(D′;(1+|x|)−γ),
γ >1, m>0 is an arbitrary integer.
3
(iii).
c (ǫ) A , as ǫ 0, ℓ,
ℓ ℓ
→ → ∀
where A :=A (α) is defined in (1.4).
ℓ ℓ
Proof. First, we prove item (i). Then we establish Lemma 2.2, and continue
with the proof of (ii) and (iii).
(i) Without loss of generality we can assume that the origin is an interior
point of the domain D. To establish (2.1) it is sufficient to show that
H :=span Ψ (s) : 0 ℓ< , s S =L2(S). (2.5)
ℓ
{ ≤ ∞ ∈ }
Suppose that there exists p L2(S), p = 0, such that p H in L2(S). Define
∈ 6 ⊥
the single-layer potential by
eik|s−y|
W(y)= p(s) ds, y R3, (2.6)
s y ∈
ZS | − |
wheredsisthesurfaceareaelement. LetU Dbeaballcenteredintheorigin.
⊂
Thenthe additiontheoremfor the fundamentalsolutionimplies that W(y)=0
for any y U.
∈
By the unique continuation principle W 0 in D. In particular W = 0 on
≡
theboundaryS. Since W isanoutgoingsolutionof( 2+k2)W =0inD′ with
∇
W = 0 on S, one concludes from the uniqueness of solutions to the Dirichlet
problem in D′ that W 0 in R3. Finally, the jump properties of the normal
≡
derivative of the single-layer potential imply that p = 0 in L2(S). We have
followed the argument from [33], p.160.
Lemma 2.2. Given g L (S), let w be the outgoing solution of the exterior
2
∈
Dirichlet problem ( 2+k2)w =0, in D′ with w=g on S. Then there exists a
∇
constant C >0, independent of w, such that
w C g , (2.7)
L2(S)
||| |||≤ k k
iwnhteegreer|,||a·n||d|:H=m||i·s||tHhlmeoc(SDo′b)o+lev||s·p||aLc2e(.D′;(1+|x|)−γ), γ >1, m>0 is an arbitrary
Proof. Let G be the Dirichlet Green’s function of the Laplacian in D′:
2+k2 G= δ(x y) in D′, G=0 on S, (2.8)
∇ − −
(cid:0) (cid:1) 2
∂G
lim ikG ds=0. (2.9)
r→∞ ∂ x −
Z|x|=r(cid:12) | | (cid:12)
(cid:12) (cid:12)
Let N be the unit normal to S poi(cid:12)nting into D(cid:12)′. By Green’s formula one has
(cid:12) (cid:12)
∂G
w(x)= g(s) (x,s)ds, x D′. (2.10)
∂N ∈
ZS
4
TheestimatefortheHm(D′)-normpartof(2.7)followsfromthisrepresentation
loc
and from the Cauchy inequality:
∂D(j)G
D(j)w(x) g x (x,s) c(x) g ,
| |≤|| ||L2(S)(cid:13) ∂N (cid:13)≤ || ||L2(S)
(cid:13) (cid:13)
(cid:13) (cid:13)
where c(x) c(d) for all x D′ su(cid:13)(cid:13)ch that the dis(cid:13)(cid:13)tance dist(x,S) d>0.
≤ ∈ ≥
For the L2-weighted norm part of (2.7) let R > 0 be such that D B =
R
x R3 : x < R . Let D′ = B D, and S be the boundary of B⊂ . The
{ ∈ | | } R R\ R R
estimate
∂G c
(x,s) , x R, (2.11)
∂N ≤ 1+ x | |≥
(cid:12) (cid:12) | |
(cid:12) (cid:12)
and formula (2.10) impl(cid:12)y (cid:12)
(cid:12) (cid:12)
w c g , (2.12)
k kL2(SR) ≤ k kL2(S)
where here and in the sequel c and C denote various constants. Also, using the
Cauchy inequality, formula (2.10), inequality (2.11) and the assumption γ >1,
one gets
1
kwkL2(|x|>R;(1+|x|)−γ) ≤ckgkL2(S) (1+ x)γ+1 ≤ckgkL2(S). (2.13)
(cid:13) | | (cid:13)L2(|x|>R)
(cid:13) (cid:13)
(cid:13) (cid:13)
To get the estimate for w L2(D(cid:13)′ ) choose R s(cid:13)uch that k2 is not a Dirichlet
eigenvalue of ∆ in D′ . Thkenk([20]R, p.189):
− R
kwkHm(DR′ ) ≤c[||(∆+k2)w||Hm−2(DR′ )+||w||Hm−0.5(SR)+||w||Hm−0.5(S)]. (2.14)
The space in the first term of the right-hand side in (2.14) is different
H
from the usual Sobolev space, but this term is equal to zero anyway because
(∆+k2)w =0.
Let m=0.5 in (2.14). Then
kwkH0.5(DR′ ) ≤c[||w||L2(SR)+||w||L2(S)]. (2.15)
Since w =g on S, then (2.12) and (2.15) imply
w L2(D′ ) c g L2(S). (2.16)
k k R ≤ || ||
Proof of Theorem 2.1, continued.
(ii) Inequality (2.3)is the sameas(2.1), sincev = u onS. Estimate(2.4)
0
−
follows from (2.3) and Lemma 2.2.
(iii) Inequality (2.3) yields the convergenceof v to v in the norm .
ǫ L2(S)
By (2.12) v v 0, as ǫ 0. On S one has v = ∞ kA·k(α)Ψ
k ǫ − kL2(SR) → → R ℓ=0 ℓ ℓ
and v = L(ǫ)c Ψ . Multiply v (R,α′) v(R,α′) by Y (α′), integrate over
ǫ ℓ=0 ℓ ℓ ǫ − ℓ P
P
5
S2 and then let ǫ 0. The result is (iii), and the proof of Theorem 2.1 is
→
completed.
ThedifferencebetweenRCandMRCis: (2.1)doesnotholdifonereplacesv
ǫ
by L A (α)Ψ ,andletsL (insteadoflettingǫ 0). Indeed,theseries
∞ ℓA=0(αℓ)Ψ dℓiverges at so→me∞points of the bounda→ry for many obstacles.
ℓP=0 ℓ ℓ
Note alsothatthe coefficients in(2.2)depend onǫ,so(2.2)is nota partialsum
P
of a series.
For the Neumann boundary condition one minimizes
∂[u + L c ψ ]
0 ℓ=0 ℓ ℓ
(cid:13) ∂N (cid:13)
(cid:13) P (cid:13)L2(S)
(cid:13) (cid:13)
(cid:13) (cid:13)
with respect to cℓ, and obt(cid:13)ains essentially the(cid:13)same results.
AccordingtoTheorem2.1thecomputationoftheoutgoingsolutionto(1.1)-
(1.2) is reduced to the approximation of the boundary values in (1.1) by the
linear combinations of the functions Ψ restricted to the boundary S. A di-
ℓ
rect implementation of the above algorithm is efficient for domains D not very
different from a circle, e.g. for an ellipse with a small eccentricity, but it fails
for more complicated regions. The numerical difficulties happen because the
spherical Hankel functions h with large values of l are bigger than h with
l l
small values of l by many orders of magnitude. A finite precision of numeri-
cal computations makes it necessary to keep the values of L not too high, e.g.
L 20. This restriction can be remedied by the following modification of the
≤
above algorithm, see [13, 16]:
Theorem 2.3. Let v := u u , where u is the solution to (1.1)-(1.2). Let
0
−
ǫ>0, and L be a nonnegative integer. Suppose U is an open subset of D.
Then there exist a finite subset z ,z ,...,z U, and the coefficients
1 2 J
{ } ⊂
c (ǫ,z ), 0 ℓ L, 1 j J = J(ǫ), such that the following inequalities
ℓ j
≤ ≤ ≤ ≤
(2.17) and (2.20) hold:
(i).
u +v ǫ, (2.17)
0 ǫ L2(S)
|| || ≤
where
J L
v (x):= c (ǫ,z )ψ (x,z ), (2.18)
ǫ ℓ j ℓ j
j=1ℓ=0
XX
and
x z
ψ (x,z)=Y (α′)h (k x z ), α′ = − , z D, x R3 D.
ℓ ℓ ℓ
| − | x z ∈ ∈ \
| − |
(2.19)
(ii).
v v ǫ (2.20)
ǫ L2(S)
k − k ≤
and
v v =O(ǫ), ǫ 0, (2.21)
ǫ
||| − ||| →
6
where
= Hm (D′)+ L2(D′;(1+|x|)−γ),
|||·||| k·k loc k·k
γ >1, m>0 is an arbitrary integer, and Hm is the Sobolev space.
Proof. (i) Note that in Theorem 2.1 we had L = L(ǫ), while now we have L
fixedandJ =J(ǫ). Butthe proofofTheorem2.3is similar to thatofTheorem
2.1. Let z ∞ be a countable dense subset of U. To establish (2.17) it is
{ j}j=1
sufficient to show that
H :=span ψ (s,z ) : 0 ℓ L, j =1,2,... =L2(S). (2.22)
ℓ j
{ ≤ ≤ }
Suppose that there exists p L2(S), p = 0 such that p H in L2(S). Define
∈ 6 ⊥
the single-layer potential by
eik|s−y|
W(y)= p(s) ds, y R3. (2.23)
s y ∈
ZS | − |
Then
W(z )= ψ (s,z )p(s) ds=0 (2.24)
j 0 j
ZS
for j =1,2,....
The continuity of the single-layer potential in R3 implies that W(y)=0 for
all y U. The rest of the proof is the same as in Theorem 2.1.
∈
Remark. Functions Ψ ∞ are linearly independent on S. Indeed, if some
{ l}ℓ=0
finite combination of these functions vanishes on S, then it also vanishes in the
exterior domain D′, since such a combinationis an outgoing solution of the ex-
teriorDirichletproblemwithzeroboundaryconditionsonS. Inparticular,such
acombinationalsovanishesonS . Sincethesphericalfunctionsareorthogonal
R
on S , it implies that such a combination must be trivial.
R
SeeSections6and7foranextensionoftheMRCmethodtostaticproblems,
and to scattering by periodic structures, respectively.
3 Iterative MRC algorithms
Let z be a point in the interior of the obstacle D, and x R3 D. Recall that
∈ \
ψ (x,z)=Y (α′)h (k x z ), (3.1)
ℓ ℓ ℓ
| − |
where h (r) are the sphericalHankelfunctions, normalizedso that h (r) eikr
ℓ ℓ ∼ r
as r + .
→ ∞
Noniterative MRC.
In this MRC implementation one chooses a set of interior points H = x
j
{ ∈
D,j =1,2,...,J, J >0 and minimizes
}
7
J L
Φ(c)= u (s)+ c ψ (s,x ) , (3.2)
0 ℓ,j ℓ j L2(S)
k k
j=1ℓ=0
XX
over c CN, where c = c . That is, the total field u(s) = u (s)+v(s)
ℓ,j 0
∈ { }
is desired to be as close to zero as possible at the boundary S, to satisfy the
required condition for soft scattering. If the resulting residual rmin = minΦ is
smaller than the prescribed tolerance ǫ, then the procedure is finished, and the
sought scattered field is
J L
v (x)= c ψ (x,x ), x D′.
ǫ ℓ,j ℓ j
∈
j=1ℓ=0
XX
If the residual rmin > ǫ then the method fails. This approach, which can be
called a Multi-point MRC, is justified by Theorem 2.3. See [13, 32, 10] for
detailsandresultsofnumericalexperiments. Theresultsshowthatthe method
isveryefficientfordomainsDofanearlysphericalshape,i.e. withoutelongated
parts. Clearly,theonlylimitationinthismethodisthecomputerresources. The
method becomes impractical for large sets of interior points H.
To remedy this situation one can use iterative MRC implementations, of
which we describe the one based on a random choice of interior points, and
another one based on an optimal choice of such points.
Iterative MRC with a random choice of points.
Informally, the Random Multi-point MRC algorithm can be described as
follows.
Fix a J > 0. Let x ,j = 1,2,...,J be a batch of points randomly chosen
j
inside the obstacle D.
Let g(s)=u (s), s S, and minimize the discrepancy
0
∈
J L
Φ(c)= g(s)+ c ψ (s,x ) (3.3)
ℓ,j ℓ j L2(S)
k k
j=1ℓ=0
XX
over c CN, where c = c . That is, the total field u(s) = g(s)+v(s)
ℓ,j
∈ { }
is desired to be as close to zero as possible at the boundary S, to satisfy the
required condition for soft scattering. If the resulting residual rmin = minΦ is
smaller than the prescribed tolerance ǫ, then the procedure is finished, and the
sought scattered field is
J L
v (x)= c ψ (x,x ), x D′,
ǫ ℓ,j ℓ j
∈
j=1ℓ=0
XX
If, on the other hand, the residual rmin > ǫ, then we continue by trying to
improveonthealreadyobtainedfitin(3.3). Adjustthefieldontheboundaryby
letting g(s):=g(s)+v (s), s S. Create another batch of J points randomly
ǫ
∈
chosen in the interior of D, and minimize (3.3) with this new g(s). Continue
8
withthe iterationsuntilthe requiredtoleranceǫonthe boundaryS isattained.
In each iteration accumulate new interior points x and the corresponding best
j
fit coefficients c . After the desired tolerance is reached, the sought scattered
ℓ,j
field v is computed anywhere in D′.
ǫ
Here is a precise description of the algorithm.
Random Multi-point MRC.
For x D, and ℓ 0 functions ψ (x,x ) are defined as in (3.1).
j ℓ j
∈ ≥
1. Initialization. Fix ǫ > 0, L 0, J > 0, N > 0. Let n = 0, and
max
≥
g(s)=u (s), s S.
0
∈
2. Iteration.
(a) Let n:=n+1. Randomly choose J points x(n) D, j =1,2,...,J.
j ∈
(b) Minimize
J L
Φ(c)= g(s)+ c ψ (s,x(n))
k ℓ,j ℓ j kL2(S)
j=1ℓ=0
XX
over c CN, where c= c .
ℓ,j
∈ { }
Let the minimum of Φ be attained at c(n) = c(n)) , j =1,2,...,J,
{ ℓ,j }
and the minimal value of Φ be rmin.
3. Stopping criterion.
(a) If rmin ǫ, then stop. Compute the approximate scattered field
≤
anywhere in D′ by
n J L
v (x):= c(k)ψ (x,x(k)), x D′. (3.4)
ǫ ℓ,j ℓ j ∈
k=1j=1ℓ=0
XXX
(b) If rmin >ǫ, and n=N , let
max
6
J L
g(s):=g(s)+ c(n)ψ (s,x(n)), x S
ℓ,j ℓ j ∈
j=1ℓ=0
XX
and repeat the iterative step (2).
(c) If rmin >ǫ, and n=N , then the procedure failed.
max
Numerical experiments based on this method are presented in the next sec-
tion. The method is relatively slow, and it can be improved by choosing the
interior points in some optimal way.
Iterative MRC with an optimal choice of points.
In this case the interior points z ,z ,... in D are chosen one at a time, and
1 2
their placement is not random. Rather, the discrepancy Ψ is minimized not
9
only with respect to the coefficients c, but also with respect to the position of
these points z .
j
Let g (s)=u (s)=u (s,α), s S.
1 0 0
∈
Minimize
L
Φ(z ,c(z )):=min min g (s)+ c ψ (s,z) , (3.5)
1 1 1 ℓ ℓ L2(S)
z∈Dc∈CNk k
ℓ=0
X
where c = c = c , L 0 is a fixed integer, and L :=
{ ℓ} { ℓm}0≤ℓ≤L,−ℓ≤m≤ℓ ≥ ℓ=0
L ℓ . Let
ℓ=0 m=−ℓ P
P P L
v (x)= c (z )ψ (x,z ), c (z )=c (z ,α). (3.6)
1 ℓ 1 ℓ 1 ℓ 1 ℓ 1
ℓ=0
X
The requirement (3.5) means that the total field u(s) = g (s)+v (s) has to
1 1
be as close to zero as possible on the boundary S, so that it approximates best
the Dirichlet boundary condition in (1.1). This is achieved by varying the in-
terior point z D and choosing the coefficients c(z) CN giving g +v the
1 1
∈ ∈
best fit to zero on the boundary S. Let the minimum in (3.5) be attained at
z D. Ifthe resultingvalueoftheresidualrmin =Φ(z ,c(z ))issmallerthan
1 1 1
∈
the prescribed tolerance ǫ, than the procedure is finished. The sought approx-
imate scattered field is v (x), x D′ (see Theorem 2.3), and the approximate
1
∈
scattering amplitude is
L
A (α′,α)=e−ikα′·z1 c (z )Y (α′). (3.7)
1 ℓ 1 ℓ
ℓ=0
X
Note that c (z )=c (z ,α).
ℓ 1 ℓ 1
The expressionfor A (α′,α) in (3.7) is obtained from (3.6) by letting x
1
| |→
in x=α′ x, because of our normalization
∞ | |
eik|x| 1
h (k x)= 1+O , x , (3.8)
ℓ
| | x x | |→∞
| | (cid:26) (cid:18)| |(cid:19)(cid:27)
and x z = x α′ z+O(1/x) as x .
| − | | |− · | | | |→∞
If, on the other hand, the residual rmin > ǫ, then we continue by trying to
improve on the already obtained fit in (3.5) as follows. Adjust the field on the
boundarybylettingg (s)=g (s)+v (s), s S,anddotheminimization(3.3)
2 1 1
∈
with g (s) instead of g (s), etc. Continue with the iterations until the required
2 1
tolerance ǫ on the boundary S is attained. At the same time keep track of
the changing approximate scattered field v (x), and the scattering amplitude
n
A (α′,α). In this construction g = u +v on S. The goal of (3.3) is to
n n+1 0 n
obtain g 0 in L2(S) as n , yielding u +v 0 in L2(S). According
n 0 n
→ → ∞ →
to Theorem2.3,this gives anapproximatescatteredsolutionv onD′ to (1.1)-
n
(1.2).
Here is a precise description of the algorithm.
MRC method with optimal choice of sources.
10